Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Write an equation that represents the statement. The sum of a number \(n\) and twice the number is 15 .

Step-by-Step Solution

Answer

The number \(n\) is 5.

Step 1: Identify the variables

From the statement given, we have a number \(n\) and the sum of this number and twice the number.

Step 2: Formulate the equation

The sum of a number \(n\) and twice the number equals 15. We can write this as: \(n + 2n = 15\). This simplifies to \(3n = 15\).

Step 3: Solve for \(n\)

Re-arrange the equation and solve for \(n\) by dividing both sides of the equation by 3. This gives us \(n = 15 / 3\)

Step 4: Simplify for \(n\)

After dividing, we find that \(n = 5\).

Key Concepts

Formulate Equations

Formulating equations is a foundational skill in algebra that involves translating words or real-life scenarios into mathematical expressions. This process requires identifying variables and understanding the relationships between them.

In our example, the statement was: The sum of a number () and twice the number is 15. To begin, we identify the variable, which is the unknown number we're trying to find, represented by (). We then move to interpret the phrase 'the sum of a number and twice the number' as an algebraic expression. Knowing that 'sum' refers to addition, and 'twice the number' means 2 times (), we can construct our equation as:
+ 2 = 15

It's crucial to translate the statement accurately to avoid any errors as we move forward to solve the equation. Remember to always look out for keywords such as 'sum,' 'product,' 'difference,' and 'quotient,' which indicate the operations needed to construct your equation.

Solve for Variable

Once the equation is formulated, the next step is to isolate the variable and solve it. In our exercise, we want to find the value of () which makes the equation true. Now that we have the simplified equation 3 = 15, solving for the variable involves undoing what has been done to the variable.

Here, () is being multiplied by 3. To isolate (), we do the inverse operation, which is division in this case. Thus, dividing both sides of the equation by 3 gives us:
= 15 / 3

Simplifying this gives us the value of (), which is 5. Solving for the variable requires a good understanding of inverse operations and always performing the same operation on both sides of the equation to maintain equality.

Simplifying Equations

Simplifying equations is a process of rewriting them in a more manageable form. This often involves combining like terms, which are terms that have the same variables raised to the same power, and performing operations to reduce the equation to its simplest form.

In the provided solution, the initial equation was + 2 = 15. The like terms here are and 2. When we combine them, we add their coefficients (the numbers in front of ) to obtain:
3 = 15

This simplification is crucial because it turns a binomial equation, which has two terms, into a monomial equation, with a single term, making it much easier to solve. The ultimate goal is to make the algebraic expression as simple as possible to facilitate the solving process. Always remember that simplifying an equation does not change its solutions, it just makes finding the solution easier.